Quantum instrument

In physics, a quantum instrument is a mathematical abstraction of a quantum measurement, capturing both the classical and quantum outputs. It combines the concepts of measurement and quantum operation.

Definition

Let $X$ be the countable set describing the outcomes of a measurement and $\{{\mathcal {E}}_{x}\}_{x\in X}$ a collection of subnormalized completely positive maps, given in such a way that the sum of all ${\mathcal {E}}_{x}$ is trace preserving, i.e. ${\textstyle \operatorname {tr} \left(\sum _{x}{\mathcal {E}}_{x}(\rho )\right)=\operatorname {tr} (\rho )}$ for all positive operators $\rho$ .

Now for describing a quantum measurement by an instrument ${\mathcal {I}}$ , the maps ${\mathcal {E}}_{x}$ are used to model the mapping from an input state $\rho$ to the outputstate of a measurement conditioned on an classical measurement outcome $x$ . Thereby the probability of measuring a specific outcome $x$ on a state $\rho$ is given by

p(x|\rho )=\operatorname {tr} ({\mathcal {E}}_{x}(\rho ))

The state after a measurent with the specific outcome $x$ is given by

\rho _{x}={\frac {{\mathcal {E}}_{x}(\rho )}{\operatorname {tr} ({\mathcal {E}}_{x}(\rho ))}}

If the measurement outcomes are recorded in a classical register, i.e. this can be modelled by a set of orthonormal projections ${\textstyle |x\rangle \langle x|\in {\mathcal {B}}(\mathbb {C} ^{|x|})}$ , the action of an instrument ${\mathcal {I}}$ is given by an channel ${\mathcal {I}}:{\mathcal {B}}({\mathcal {H}}_{1})\rightarrow {\mathcal {B}}({\mathcal {H}}_{2})\otimes {\mathcal {B}}(\mathbb {C} ^{|x|})$ with

{\mathcal {I}}(\rho ):=\sum _{x}{\mathcal {E}}_{x}\left(\rho \right)\otimes \vert x\rangle \langle x|

Here ${\mathcal {H}}_{1}$ and ${\mathcal {H}}_{2}$ are the Hilbert spaces corresponding to the input and the output quantum system of a measurement.

A quantum instrument is an example of a quantum operation in which an "outcome" $x$ of which operator acted on the state is recorded in a classical register. An expanded development of quantum instruments is given in quantum channel.

References

E. Davies, J. Lewis. An operational approach to quantum probability, Comm. Math. Phys., vol. 17, pp. 239–260, 1970.
Distillation of secret key paper
Another paper which uses the concept

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